Stability of small regulatory motifs may contribute to biological network organization

where the variable xi represents the state of the ith node and fi represents the combined influence of all nodes having connections with the ith node. The fi may be linear or nonlinear functions. The local stability properties of the system about its (possibly multiple) equilibria can be determined using Lyapunov’s indirect method (Khalil, 2002). This involves determining the location of the eigenvalues of the Jacobian matrix, J = {ai j}= {∂ fi/∂x j}, evaluated at the equilibrium of interest. The terms ai j represent the sign and strength of influence of the jth node onto the ith node. If this term is zero, the jth node does not influence the ith node at this equilibrium. Thus the Jacobian matrix serves to denote the local connectivity of the system. It can be reduced to the corresponding adjacency matrix by normalizing the ai j to ones or zeros. In this study it is assumed that the self-connections for all nodes of the motifs, aii (the diagonal terms of the Jacobian matrix) are always negative. This assumption reflects the commonly observed mechanisms of constitutive degradation or inactivation of the biological entities, including gene products, phosphorylation states of signaling molecules or depolarization states of neurons. Further assumptions about the values of ai j adopted in computational analysis are described below.